The final report from a workshop on Morse theory in low-dimensional and symplectic topology includes the following question, posed by Michael Hutchings: *Can we apply symplectic geometry to solve the Schönflies conjecture? Can we deform any $S^3$ to be pseudo-convex?*

My question is simply about interpreting Hutchings' latter "deformation" question and its connection to the Schoenflies conjecture. Suppose $P \subset S^4$ is a smoothly embedded $S^3$, which we know bounds a 4-manifold $X$ homeomorphic to $B^4$ by a theorem of Brown. Viewing $P$ inside $\mathbb{C}^2 = S^4-\{pt\}$, is Hutchings asking whether we can perturb $P$ so that it is the pseudoconvex boundary of a Stein domain?

A positive answer would imply (the smooth 4-dimensional) Schoenflies conjecture, right? First, a Stein-fillable (or weakly/strongly symplectically fillable, for that matter) contact structure on $S^3$ must be tight (Eliashberg-Gromov), hence isotopic to $(S^3,\xi_{\operatorname{std}})$. Now since a Stein filling of $(S^3,\xi_{\operatorname{std}})$ must be diffeomorphic to $B^4$ (Eliashberg), we conclude that $P$ bounds a smoothly standard $B^4$.

Finally, this seems to suggest a (perhaps) weaker question: *Can we perturb $P$ and endow it with a contact structure making $X$ into a weak symplectic filling?* A positive answer again seems to imply Schoenflies. Indeed, the above arguments would imply that $P$ is a standard contact $S^3$. Note that $X$ is homeomorphic to $B^4$, so it is certainly symplectically aspherical for any smooth and symplectic structure. By Eliashberg (perhaps Eliashberg-McDuff?), a symplectically aspherical filling of $(S^3,\xi_{\operatorname{std}})$ is diffeomorphic to $B^4$.

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